Question 2 Solve the polynomial inequality: \( x^{3}(x-6)^{2}(x+3)<0 \) State your answer using interval notation. Use \( U \) for union and oo for \( \infty \). .

To solve the polynomial inequality \( x^{3}(x-6)^{2}(x+3)<0 \), first identify the roots of the polynomial: \( x = 0 \), \( x = 6 \), and \( x = -3 \). Now, we can determine the sign of the product in the intervals defined by these roots: \( (-\infty, -3) \), \( (-3, 0) \), \( (0, 6) \), and \( (6, \infty) \). - In \( (-\infty, -3) \): pick \( x = -4 \) ⇒ the product is negative. - In \( (-3, 0) \): pick \( x = -1 \) ⇒ the product is positive. - In \( (0, 6) \): pick \( x = 1 \) ⇒ the product is negative. - In \( (6, \infty) \): pick \( x = 7 \) ⇒ the product is positive. The inequality is satisfied where the product is negative: \( (-\infty, -3) \) and \( (0, 6) \). Also, the root \( x = -3 \) makes the product zero (not negative), while \( x = 0 \) and \( x = 6 \) also yield zero for the polynomial, so they are not included in the solution. Thus, the final answer in interval notation is: \[ (-\infty, -3) \cup (0, 6) \]